Use the critical value method to solve each rational inequality.Write each solution set in interval notation.

Question

Quizplus · Accepted Answer

For D, we first need to find the critical values by setting the denominator equal to zero:
$x-1=0$ and $x+3=0$.
This gives us critical values of $x=1$ and $x=-3$.
Next, we need to test the intervals defined by these critical values by selecting test values in each interval for which the inequality can be easily evaluated.
For example, let's test the interval $(-\infty,-3)$ by selecting a test point $x=-4$:
$\frac{(x-2)(x-4)}{(x-1)(x+3)}>0$
$\frac{(-6)(-8)}{(-5)(1)}>0$
$\frac{48}{5}>0$
Since the inequality is true for this test value, we know that the solution set includes the interval $(-\infty,-3)$.
Similarly, testing the interval $(-3,1)$ with a test point of $x=0$ and the interval $(1,\infty)$ with a test point of $x=2$ gives us the solution set:
$(-\infty,-3)\cup (1,\infty)$.

Use the Critical Value Method to Solve Each Rational Inequality